Use Mathematica (REQUIRED) and ... on the interval between 0 and a, thus $0 < x < a$, consider the functions $\phi(x, n) = \sin(\frac{n\pi x}{a})$ for different values of $n \in \mathbb{N}$. (the constant a is not important - except that it has the same dimension as x; if you need a numerical value, you may set it to 1).
a. [1] Graph functions $\phi(x, n)$ for n=1-5 — all 5 graphs into one single diagram!
b. [1] Evaluate the 5 integrals $\int_0^a \phi(x, n) \cdot \phi(x, n) dx$ (note: same n) for n=1-5. 5 lines of commands.
c. [1] Evaluate the 20 integrals $\int_0^a \phi(x, n) \cdot \phi(x, m) dx$ ($n, m = 1-5, n \neq m$).